The first six terms in the right-hand side of Eq. (REF ) represent a renormalization of the quantum kinetic energies (the terms \({\hbar ^2}|\nabla \psi _\alpha |^2/{2m}\) ) of the fluids due to the momentum-coupling.
The presence of those terms is necessary for the Galilean invariance of the internal fluid energy.
The rest four terms of Eq. (REF ) are equivalent to the ones used in equation (5) of [1]}.
The last four terms in Eq. (REF ) were also used in equation (3) of [2]}, however, the relative signs are different from the corresponding signs in equation (5) of [1]}.
This is not surprising because the calculations are focused on the momentum-coupling in [1]}, [5]}, and on the gradient-coupling in [2]}, [7]}.
The "coefficients" by the terms of the form \((\psi _\alpha \nabla \psi _\alpha ^*)(\psi _\beta \nabla \psi _\beta ^*)\)  in Eq. (REF ) are seen to be non-trivial functionals of the SF densities, which is a consequence of the representation form via the wave function gradient.
However, once the whole expression is written in terms of the basic degrees of freedom as explained above, the perturbative structure of the excitation energy in the framework of the effective field theory becomes obvious.
